Proof that G is an Abelian Group if f(a) = a^(-1) is a Homomorphism

In this video I prove that G is an abelian group if f(a) = a^(-1) is a group homomorphism. This problem is from a book called "Foundations of Higher Mathematics" and it was written by Fletcher and Patty. This is a good book for learning to write proofs.

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Thank you:)

[BrianCarloRivera]

I like these kinds of videos math sorcerer. please make a playlist regarding you proving stuff from advanced calculus. From Royden

[demongeminix]

I used this book back in college in 2004 for a Math History class. I still have it.

If you're going to get it, get it used. The price for it new is outrageous.

[josesilesramirez]

Thanks so much for this video. I will say that it is much better when you show your organic thinking and scratch work instead of presenting a clean well organized proof. It is better because we learn not just the results but also thinking strategies.

[bengtbengt]

Nice to see some proofs from group theory. I hope to see more soon!

[eflat]

These problems you work out are still above my skill level, but I still watch them hoping to still learn something.:-)

[dreimikemount]

Brought me back memories of a 1-semester study-unit "Groups & Vector Spaces" from the 1st year of my BSc Maths degree.

[malawigw]

I have discovered a truly marvelous proof of this, which this comment box is too small to contain

[acdude]

Nice problem to demonstrate basic group ideas, introducing homomorphisms in a proof using inverses of elements.

Gallian, Durbin, and Hernstein (out of print) are good intro books.

[mimicoolll]

i just gotta say you’re one of my favorite youtubers ever!

[Kavyakavya-srrd]

I love maths and I'm mathematics ug student ... I fear about Abstract algebra but I love Maths... After ur video I will definitely fall in fearles love with abstract algebra.. thank you sir💜✨.. Keep doing these kind of playlist Sir 💜🥺

[kundananji.simutenda]

Proff requesting if you can make more videos on mathematical statistics.

[God-ldll]

wondering why it isn't more often called a communitive group being autological. If i am correct on that.

[tamptus]

We can see more: If G is not abelian and f:G->H and multiplication in H is swapped ( x (*H) y :-= y (*G) x ). Then f(a) = a^(-1)
is an isomorphism.
Is swapping allways isomorphic? No! There is a Loop (Quasigroup with neutral) with 5 Elements and swapping is not isomorh.

[WitchidWitchid]

Brings me back to my Group Theory days. I enjoyed figuring out those.types.of.proofs. I always liked to think about alternate ways to prove a theorem I. Except when I got stumped. Then I was just happy to discover any proof that would work correctly and at that point I would leave well enough alone...LOL.

[MrMegatherium]

In this problem, can we assume f has an inverse? I used it in my proof, but I am not sure if it's right.

[kilian]

You can also show the other direction, that if G is abelian then it is a homomorphism.